What does the "Absolute" setting in the Math node do?
The Absolute mode is similar to the mathematics $\left\vert X \right\vert$ operator. Any negative value is turned positive.

The 2nd input on the node is meaningless and has no effect at all (thanks Carlo for checking this out in Blender's source code).

Below is an example that illustrates the mode. The image has values in the interval $[0, 1]$ (black to white). The first math node multiplies everything by $-1$, thus turning the interval to $[-1, 0]$. Everything that equals or is smaller than $0$ is rendered black, so the image turns black.
When you plug this into the absolute node, all negative numbers are turned positive, and the image is displayed again.